In mathematics, a layer group is a three-dimensional extension of a wallpaper group, with reflections in the third dimension. It is a space group with a two-dimensional lattice, meaning that it is symmetric over repeats in the two lattice directions. The symmetry group at each lattice point is an axial crystallographic point group with the main axis being perpendicular to the lattice plane.
Table of the 80 layer groups, organized by crystal system or lattice type, and by their point groups
Triclinic1 p12 p Monoclinic/inclined3 p1124 p11m5 p11a6 p112/m7 p112/a Monoclinic/orthogonal8 p2119 p211110 c21111 pm1112 pb1113 cm1114 p2/m1115 p21/m1116 p2/b1117 p21/b1118 c2/m11 Orthorhombic19 p22220 p212221 p2121222 c22223 pmm224 pma225 pba226 cmm227 pm2m28 pm21b29 pb21m30 pb2b31 pm2a32 pm21n33 pb21a34 pb2n35 cm2m36 cm2e37 pmmm38 pmaa39 pban40 pmam41 pmma42 pman43 pbaa44 pbam45 pbma46 pmmn47 cmmm48 cmme Tetragonal49 p450 p51 p4/m52 p4/n53 p42254 p421255 p4mm56 p4bm57 p2m58 p21m59 pm260 pb261 p4/mmm62 p4/nbm63 p4/mbm64 p4/nmm Trigonal65 p366 p67 p31268 p32169 p3m170 p31m71 p1m72 pm1 Hexagonal73 p674 p75 p6/m76 p62277 p6mm78 pm279 p2m80 p6/mmm
Correspondence with plane groups The surjective mapping from a layer group to a wallpaper group (plane group) can be obtained by disregarding symmetry elements along the stacking direction, typically denoted as the z-axis, and aligning the remaining elements with those of the plane groups. The resulting surjective mapping provides a direct correspondence between layer groups and plane groups (wallpaper groups).
+ Surjective mapping from Layer Groups to Plane Groups # Layer Group # Plane Group 1 p1 1 p1 2 p 2 p2 3 p112 2 p2 4 p11m 1 p1 5 p11a 1 p1 6 p112/m 2 p2 7 p112/a 2 p2 8 p211 3 pm 9 p211 4 pg 10 c211 5 cm 11 pm11 3 pm 12 pb11 4 pg 13 cm11 5 cm 14 p2/m11 6 p2mm 15 p2/m11 7 p2mg 16 p2/b11 7 p2mg 17 p2/b11 8 p2gg 18 c2/m11 9 c2mm 19 p222 6 p2mm 20 p222 7 p2mg 21 p222 8 p2gg 22 c222 9 c2mm 23 pmm2 6 p2mm 24 pma2 7 p2mg 25 pba2 8 p2gg 26 cmm2 9 c2mm 27 pm2m 3 pm 28 pm2b 3 pm 29 pb2m 4 pg 30 pb2b 3 pm 31 pm2a 3 pm 32 pm2n 4 pg 33 pb2a 4 pg 34 pb2n 5 cm 35 cm2m 5 cm 36 cm2e 3 pm 37 pmmm 6 p2mm 38 pmaa 6 p2mm 39 pban 10 p4 40 pmam 7 p2mg 41 pmma 6 p2mm 42 pman 9 c2mm 43 pbaa 7 p2mg 44 pbam 8 p2gg 45 pbma 7 p2mg 46 pmmn 10 p4 47 cmmm 9 c2mm 48 cmme 6 p2mm 49 p4 10 p4 50 p 10 p4 51 p4/m 10 p4 52 p4/n 12 p4gm 53 p422 11 p4mm 54 p422 12 p4gm 55 p4mm 11 p4mm 56 p4bm 12 p4gm 57 p2m 11 p4mm 58 p2m 12 p4gm 59 pm2 11 p4mm 60 pb2 12 p4gm 61 p4/mmm 11 p4mm 62 p4/nbm 11 p4mm 63 p4/mbm 12 p4gm 64 p4/nmm 11 p4mm 65 p3 13 p3 66 p 16 p6 67 p312 14 p3m1 68 p321 15 p31m 69 p3m1 14 p3m1 70 p31m 15 p31m 71 p1m 17 p6mm 72 pm1 17 p6mm 73 p6 16 p6 74 p 13 p3 75 p6/m 16 p6 76 p622 17 p6mm 77 p6mm 17 p6mm 78 pm2 14 p3m1 79 p2m 15 p31m 80 p6/mmm 17 p6mm
See also
Point group Crystallographic point group Space group Rod group Frieze group Wallpaper group
References
External links Bilbao Crystallographic Server, under "Subperiodic Groups: Layer, Rod and Frieze Groups" Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin CVM 1.1: Vibrating Wallpaper by Frank Farris. He constructs layer groups from wallpaper groups using negating isometries.